One of the things I've always enjoying about talking with you has been your willingness to engage in debate on good faith. So, other than the back-handed compliment, did you have a point to make?
Not one I can make here. I've already made my point I think as sharply as it can be made. Like Umbran, I'm handicapped by the fact that you've veered off from discussing how politics impacts your gaming, into discussing politics period. This is hardly surprising though, since you've made it clear that for you all discussions are political and there is no such thing as apolitical discussion. Unlike some, my politics aren't privileged here, so I must stop following this particular thesis around.
I do want to take up a different one, but before I do, my critique of your position does not depend on you having made a logical error or conflating the meaning of the word 'politics'. I consider your position logically coherent, but I'm highly critical of your position anyway for reasons that I've been able to only suggest - and which you confirmed.
That out of the way, you've now advanced this bit in your defense which is completely wrong:
They are conflated; we wouldn't use the exact same word to describe all of them if they weren't.
Unfortunately, the English language is not nearly so regular or precise, which is why one doesn't try to write computer code in natural langauge. We often use multiple words with the same spelling and sound but wildly different meanings in different contexts. We do this even in the case of words which have not yet been given a thorough Newspeak revision to make them mean what certain groups would prefer that they mean, rather than what they have meant.
A good example of a word that had multiple confusing and different meanings even before the Ministry of Truth got a hold of it, is the word 'liberal'.
This word means:
1) 'charitable' - A man with a 'liberal' spirit is a charitable man who gives away generously.
2) 'a large quantity' - But a man can also put a liberal amount of pepper on his pasta, which simply means 'a lot' and not 'charitable'.
3) 'useless' - We see this use of the word 'liberal' in the phrase 'liberal arts'. Liberal in this sense means, "without practical utility", and is employed in contrast with practical arts like architecture, glass blowing, painting, masonry, carpentry, and engineering. The liberal arts are the ones without practical value, as opposed to the ones learned by 'mere' tradesmen. A liberal arts college is the sort, historically, that teaches leisured members of the upper class the things that they are expected to know, as opposed to mere professional skills.
4) 'receptive to change' - This is one of the several varying meanings of 'liberal' in the context of politics, where it refers to a man who by inclination when they see that the status quo has defects, believe that he burden of proof lies on him who wishes to maintain the status quo rather than on him who is desirous of change. They readily embrace new and different policies, hoping for improvement over the existing situation. This is in contrast to 'conservative', who - whether he agrees a problem exists or not - is a man who by inclination would rather see proof that the particular changes that are desired will bring improvement.
5) 'a philosophy that believing in the value and goodness of private property' - This is the original meaning of the word 'Liberal' with respect to politics, and describes the philosophy of thinkers like John Locke, Adam Smith, and others who felt that ownership of property was not only a fundamental human right, but the most critical and important human right. This meaning still is used occasionally in Europe to refer to 'Liberal Economics', some of the 'liberal' parties of Europe are thus pro Free Market parties, but is now so rare in American usage that people who mean this by the word 'Liberal' are forced to prepend the term with 'Classical' to make it clear what they mean.
6) 'figurative' - As applied to speech, logic, or law the word 'liberal' means to take a non-literal approach to meaning and to seek to find a basis of understanding not based on the literal meaning of the words, but for example on the basis of how the interpretation would achieve some desired goal. For example, in the context of RPGs, a liberal DM would be the opposite a DM that applies the letter of the law, but rather uses fiat and interpretation to arrive an interpretation that best serves the goals of the game (for example, what would be most 'fun for everyone').
7) 'politically left-wing' - Generally speaking, this is a term without any definite meaning because it's always relative to the prevailing norms, but often in the Western context means 'opposed to the concept of private property to one degree or the other'.
8) 'broad-minded' - A liberal minded man is one that is undogmatic in his beliefs. He's inclined to be tolerant of positions that differ from his own, and generally accepts the notion of private thought, belief, and action as being as important and valid as the liberal philosopher accepts the value of private property.
These terms are linked sometimes entirely by accident, since they first arose as metaphors for the sort of virtues that ought to grace a 'free' man - by which the original people who were using the phase meant a member of an entitled inherited aristocracy - or as a result of trying to define what it meant to be 'free'. As they exist to day, the words 'liberal' refers at a whole bunch of frequently directly contrasting ideas, as the word has been widely misapplied by those that want to bask in the esteem the word once enjoyed.
And that's before we get into the Newspeak wringer that this word and others have been run through, particularly in recent decades. I strongly encourage people to read dictionaries of the American language from their introduction on, skipping ever few decades to see how a word has been redefined by the lexicographists either because its meaning has changed over time or because their perception of the idea has changed over time as their own biases and beliefs as a community shifted. It's one of the most fascinating and important studies one can do, as very quickly if you do side by side comparison of dictionaries and encyclopedias you'll quickly realize that certain words mean entirely different things in the mouths of some speakers than they do in the mouths of others.
Interesting case in point, for the interested observer: "What does the word 'politics' mean to Gradine?"